When I completed taxes in turbotax, I thought it should recalculate
and I should get refund from MA and pay RI state taxes instead.
But turbotax think that I should pay MA taxes, not RI.
Since you live in Rhode Island I went to the RI Government website:
Frequently Asked Questions Regarding 2011 Rhode Island Personal Income
Tax Withholding and Revised Personal Income Tax Rates
7. I am a Rhode Island resident, but I work in another state. Is my employer required to withhold Rhode Island income tax from my wages?
An employee who is a resident of Rhode Island is subject to Rhode
Island withholding for wages earned in Rhode Island. If a Rhode
Island resident works in another state, such as Massachusetts or
Connecticut, the wages will be taxed in the state where the wages are
earned. The Rhode Island resident may claim a credit for income tax
paid to the other state on the RI-1040 resident income tax return.
So If the State of Massachusetts Taxes are greater than the taxes that RI was expecting. Then when RI credits you for what you paid MA then the Income tax you should pay RI will be zero.
Of course all this depends on other income you may have that isn't MA income. And is income tax only.